This leads us to think that there is some sort of miracle involved in arriving at the fridge to get a beer. — Ludwig V

There's no miracle. Motion isn't continuous; it's discrete.

$0, 1, 0, ..., 1\\0, 1, 0, ..., 0$

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off.

As a comparison, even though imaginary numbers have a use in mathematics it is more than just physically impossible for me to have $\sqrt{\text{-}1}$ apples in my fridge; it is metaphysically impossible.

No pretend physics can allow for me to have an imaginary number of apples in my fridge and no pretend physics can allow for the above two mathematical sequences to model the state of a lamp over time.

With Thomson's lamp, these are our premises:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t₀
P5. The button is pushed at least once between t₀ and t₁
P6. The lamp is either on or off at t₁

And these are our conclusions:

C1. If the lamp is on at t₁ then the button was pushed to turn it on, prior to which it was off
C2. If the lamp is off at t₁ then the button was pushed to turn it off, prior to which it was on
C3. The button is pushed $n \in \mathbb{N}_1$ times between t₀ and t₁

These conclusions prove that a supertask is not performed.

The basic confusion is not understanding that an infinite sequence has no end. — fishfry

I understand that it has no end. That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes.

And that’s precisely why the question of whether the lamp will be on or off at two minutes will never present itself. — Fire Ologist

It does present itself because the lamp must be either on or off after two minutes as per the law of excluded middle.

We want to know what would happen to the lamp if we were to push the button to turn it on and off at successively halved intervals of time within two minutes. Nothing else happens to the lamp except what we cause to happen to it by pushing the button.

Is it left on or off after we stop pushing the button? If you can’t answer this then you must accept that supertasks are metaphysically impossible.

How is that? How is it on or off at or after two minutes? — Fire Ologist

Because it's a lamp. If it exists at 12:02 then it's either on or off, and it exists at 12:02.

It cannot be a function of a switch that operates by switching every half of the prior interval. — Fire Ologist

And that's precisely why supertasks are impossible.

We are imagining, for the sake of argument, that we are in some alternate universe with physical laws that allow us push a button to turn the lamp on and off at successively halved intervals of time. We want to know what happens to the lamp if we do this (and only this).

As you say, we don't get to "just [assume] something exterior to the premise about time and lamps."

So the only premises we are allowed to work with are:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t₀
P4. The lamp is either on or off at t₁
P5. The button is pushed at successively halved intervals of time between t₀ and t₁

So we ask: is the lamp on or off at t₁?

If you cannot provide a coherent answer then you must accept that the premises are inconsistent. Given that there's nothing problematic about P1 - P4, we must accept that the problem lies with P5. It is necessarily false. Supertasks cannot be performed.

Because the switch is not designed to ever present the question. — Fire Ologist

I don't know what you mean by this.

Given that the lamp must be either on or off after two minutes we must ask the question. If you cannot provide a coherent answer then you must accept that your premise – that the lamp has turned on and off an infinite number of times – is necessarily false, and so that supertasks are metaphysically impossible.

Or more precisely, not designed to function at or after two minutes. — Fire Ologist

That's also true about the first two scenarios – neither switches after 1 minute and 30 seconds – and yet we can still answer the question about the lamp after 2 minutes, and even after 2 years.

The lamp is off. It turns on and off only as described:

Scenario 1
The lamp turns on after 1 minute.

Is the lamp on or off after 2 minutes? It's on.

Scenario 2
The lamp turns on after 1 minute and off after a further 30 seconds.

Is the lamp on or off after 2 minutes? It's off.

Scenario 3
The lamp turns on after 1 minute, off after a further 30 seconds, and so on ad infinitum.

Is the lamp on or off after 2 minutes?

In all three scenarios the switch is "designed to function within two minutes."

The lamp is off. After one minute the lamp turns on. Is the lamp on or off after two minutes? It's on because it was turned on after one minute and then never turned off again.

If you can't apply the same reasoning to the lamp having turned on and off an infinite number of times before the end of two minutes then you must accept that it makes no sense for the lamp to have turned on and off an infinite number of times before the end of two minutes.

I don’t understand. How do you ever arrive at the two minute mark? — Fire Ologist

I don't know what you mean by "arrive" at the two minute mark. Two minutes just pass. That's how the world works.

Imagine I am facing a clock with my back to a lamp. The experiment starts at 12:00. Through some automated process the lamp turns on and off at successively halved intervals of time. When the clock shows 12:02 I turn around. Is the lamp on or off?

That means that any answer whatever is equally valid — Ludwig V

That's not true, as I explained here, and as I alluded to above. It is not just the case that whether the lamp is on or off after two minutes is undefined but that the lamp cannot be either on or off after two minutes.

As Thomson says in his paper, "the impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be associated arithmetical sequence is convergent or divergent."

Now if I can just get Michael to agree! — fishfry

I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge.

I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. See my previous post and my initial defence of Thomson on page 13.

But which is not defined. — Ludwig V

It's more than that; the lamp can't be on and can't be off, even though it must be one or the other. This is a contradiction, and so therefore the supertask is proven impossible in principle.

I'm not at all clear how the ordinary logic of cause and effect would apply in the context of hypothetical physical laws. But we are clearly not dealing with the ordinary physical world, and that leaves us free to imagine anything at all. — Ludwig V

If it's on at t₁ then either it was left on before t₁ or it was left off before t₁ and then turned on at t₁.

This is a straightforward logical point that does not depend on what the physical laws are.

To make it very simple, Thomson's lamp proves that these premises are logically inconsistent:

P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t₀
P5. The button is pushed at successively halved intervals of time between t₀ and t₁
P6. The lamp is either on or off at t₁

A supertask is not simply an infinite sequence of numbers.

In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect.

It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause – even if it's magic.

If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off.

Given this, if the lamp is on at t₁ then either:

a) it was turned and left on prior to t₁, or
b) it was turned and left off prior to t₁ and then turned on at t₁

But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false. Therefore the lamp is not on at t₁. Similar reasoning shows that the lamp is not off at t₁ either.

But it's not my only solution. I've said (several times) that "Lamp is on" and "Lamp is off" are also valid solutions. — fishfry

The lamp is on only if the button was pushed to turn it on, prior to which the lamp was off. Even if you want to introduce magic it is on only if magic turned it on, prior to which the lamp was off.

So if you want to say that the lamp is on after two minutes then you must accept that at some final time prior to two minutes the lamp was off. This is a simple logic fact.

Except the supertask doesn't allow for this. So, as Thomson argues, the lamp cannot be on after two minutes. And for the same reasoning cannot be off after two minutes. This is a contradiction and so the supertask is proven impossible in principle.

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

We stop at the single issue being discussed: performing some action at arbitrarily small intervals of time. So taking the code here we assume that each line is run in an instant with the exception of the wait i *= 0.5 line which waits for the specified time in seconds. The logic of the code still behaves exactly as we would expect. Thomson is asking us what is output when echo isLampOn runs.

If your only solution is to insert the line isLampOn = 'a plate of spaghetti' after while (true) { ... } and before echo isLampOn then you are not answering the question as posed.

If you cannot make sense of the echo isLampOn line without inserting some arbitrary code before it then you must accept that it doesn't make sense for while (true) { ... } to complete. The arbitrary code you are trying to insert is a smokescreen to disguise this impossibility, exactly like your magic turning the lamp into a plate of spaghetti.

you haven't demonstrated any contradictions in TL — SophistiCat

Thomson does that himself in his paper. I am defending his paper and explaining why Benacerraf's response to it fails. See here where I first brought it up.

Perhaps you could explain which part of my (or Thomson's) reasoning you reject? You're a coder so perhaps you could even address the code here. Simply saying "it's wrong" is hardly a meaningful criticism. If that's all you have to say then I will simply reply with "it's right".

nor linked it to continuous motion. — SophistiCat

I did so in the post yesterday. Just as if we push a button an infinite number of times within two minutes the lamp can neither be on nor off after two minutes, if we run through an infinite succession of sensors when running a mile the lamp can neither be on nor off after finishing the run.

Given that the lamp must be either on or off, this is a contradiction, and so therefore it is proved that one cannot have run through an infinite succession of sensors.

You are just restating - reimagining - Thompson's Lamp thought experiment, which has nothing to do with continuous motion as such — SophistiCat

I’m using Thomson’s lamp to show that continuous motion entails contradictions.

and repeating once more your baseless conclusion — SophistiCat

It’s not baseless. I’ve explained it quite clearly here and here and in many other comments.

The problem is that if motion is continuous and if the sensors are set up as stated then the lamp can neither be on nor off after the run is completed, which is a contradiction.

One or more of the premises is necessarily false. So either motion is not continuous or we cannot set up sensors at an infinite succession of halfway points. The latter would seem to suggest that there aren’t an infinite succession of halfway points and so would entail the former anyway.

Unlike Zeno's thought experiments, which deal with examples of ordinary motion — SophistiCat

Continuous motion suffers from the same problem. We can imagine sensors at each successive half way point that when passed turn a lamp on or off. Is the lamp on or off when we finish our run?

The simple solution is to say that motion isn’t continuous. Discrete motion at some scale is a metaphysical necessity.

Quantum gravity and quantum spacetime are viable theories so it’s not a hard pill to swallow.

So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. — fishfry

A supertask is not simply an infinite sequence.

With a supertask we are given some activity to perform and we assume that it is physically possible to perform this activity at successively halved intervals of time. We are then asked about the causal consequence of having done so.

We do not get to introduce additional (and nonsensical) premises such as "and then the lamp magically turns into a plate of spaghetti, prior to which the lamp was neither on nor off."

The lamp must be either on or off after two minutes. If the lamp is on after two minutes then it is on only because the button was pushed to turn it on, prior to which the lamp was off. If the lamp is off after two minutes then it is off only because the button was pushed to turn it off, prior to which the lamp was on. The supertask doesn’t allow for either of these scenarios and so is proven impossible in principle.

I think you'll find that's because it makes no sense to answer the question.

In other words, it also makes no sense to answer the question with "on" or "off". — Ludwig V

The lamp is either on or off at t₁. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times.

Manhattan voted 85% for Joe Biden, and registered Democrats outnumber Republicans eight to one in New York. The Biden/Harris campaign and a whole host of anti-Trump Democrats pay the judge's daughter an obscene amount of money to work for them. A simple change of venue would have been an appropriate fix. — NOS4A2

It's a New York crime so was always going to be tried in New York.

If it makes you feel better, his Florida trial will be in an area that heavily favours Republicans, so you can be grateful of a biased jury in his favour, and with a biased judge he appointed.

I suppose prosecutors would have had to prove that Trump first new about this law, and then intended to violate it. — NOS4A2

Ignorantia juris non excusat.

Well, in the way philosophy pictures them yes. I moved the discussion here because the article above provides some history of the parallel picture that neuroscience labors under. Philosophy has never liked being wrong so the fact that we can be (and that we are responsible for that) leads it to create the conclusion that we must not have direct access to the world (or we are ensured it), that we only see the “appearance” of something, or that our individual perspective is somehow partial or lacking or individual (my “sensation” or “perception”). — Antony Nickles

There are plenty of good reasons, supported by science, to believe indirect realism over direct realism, as I discussed at length here.

But I don't understand how we got to this point. You were saying something about us wanting to help each other if we're in pain, and somehow conclude from this that indirect realism is false? Your reasoning is confusing.

Michael, This post may be of interest to you. — fishfry

I'm afraid it's not, because it doesn't address the issue of supertasks.

For supertasks, we have this:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t₀
P4. The lamp is either on or off at t₁

Q. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

It makes no sense to answer this question with "a plate of spaghetti" or "$1\over2$".

Have you even read Thomson's paper? This is the most relevant part:

There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. So if the lamp was originally off, and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on, according to Russell's recipe. After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

The answer to his question isn't "a plate of spaghetti", it isn't "0", it isn't "1", and it isn't "1/2".

The only coherent answers are "on" and "off" – except as he argues, it can't be either, and so therefore it is impossible, even in principle, to have succeeded in pressing the button an infinite number of times.

I think the judge and jury were partial — NOS4A2

Perhaps. Trump is a divisive figure. Most people either hate him or worship him. But if there's evidence of crimes then he still needs to be prosecuted. How would you go about finding an impartial jury, and what makes you think that this wasn't already done in this case?

the crime was made up — NOS4A2

I don’t really know what this means. He was prosecuted under 175.10 - Falsifying Business Records In the First Degree, with the intent to violate 17-152 - Conspiracy to Promote or Prevent Election.

the conviction was bought and sold — NOS4A2

I don't really know what this means either. Are you suggesting that the jury were paid to find him guilty?

Biden’s Banana Republic prevails. Trump is now a Mandela, and they destroyed the justice system to rig another election. — NOS4A2

You think the jury was planted?

The code here is effectively the same as a recursive function.

My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V

I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible.

The ideas of consciousness, sensation, appearance, reality, are all manufactured by philosophy, partly to feel like we are necessarily special, as I discussed above. — Antony Nickles

What do you mean by saying that they're manufactured? Are you saying that they're a fiction? You need to prove that, not simply assert it – and so we're doing philosophy.

The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined. — Ludwig V

That's precisely why supertasks are impossible.

The lamp is off at t₀. The lamp must be either on or off at t₁. But if the button has been pushed an infinite number of times between t₀ and t₁ then the lamp can neither be on nor off at t₁.

It's strange that some are taking the very thing that proves that supertasks are impossible as proof that the proof isn't a proof.

Witt would be showing how this “problem” and ontology are manufactured by our human desires. — Antony Nickles

So because we only care about aspirin when we have a headache then it follows that first person private sensations don't exist, or that if they do exist then they are the same for all people?

That's obviously a non sequitur.

Maybe the way to put this is that equating our pains is not how pain is important to us. If this situation actually did happen, what would matter to us about comparing pains would be attending to one or other of us. Philosophy abstracts this discussion to a place of equating pains, and then creates “sensation” as a kind of object, rather than just me expressing how I feel (which is too vague), so that knowledge might stand in the place of our having to react to someone in pain. What it wants is to be sure of the other person (and what to do), and not have to make the leap of faith of treating them as a person in pain. — Antony Nickles

When we're discussing something like the hard problem of consciousness and the ontology of sensations then it very much matters to us if our pains are the same or not.

All you seem to be arguing is that when we're hungover after a night of heavy drinking then we should care more about whether or not there is some aspirin. I doubt anyone disagrees. But I fail to see the relevance of this on a philosophy forum.

Right, but this might be because one is feigning agreement because they are pitying the other, or being stoic, and maybe not some way for our pain to be “truly” the same, which philosophy perhaps simple creates in order to impose the requirement we wanted all along. — Antony Nickles

Or it's because the sensation I have when I stab myself in the arm is unlike the sensation you have when you stab yourself in the arm, and so our pains are not the same and we don't know one another's pain.

As a matter of connection and to identify with the other person, we say our pain is the same, that we know the other’s pain. — Antony Nickles

That we say it isn't that it's true.

Again, Witt’s point is not to be right — Antony Nickles

Then I will simply say that Wittgenstein is wrong and so we shouldn't listen to what he has to say.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises — sime

I think it does. We need to examine the process in reverse, and remember that the lamp is on iff the lamp was off and the button was pushed to turn it on. We're discussing a supertask after all, not simply the infinite sequence {off, on, off, ...}.

If the lamp is on at t₁ then either:

a) the button was pushed to turn the lamp on before t₁ and then it was left on until t₁, or
b) the button was pushed to turn the lamp off before t₁ and then it was left off until t₁ when the button is pushed to turn it on

Neither (a) nor (b) are possible given the defined supertask – the lamp is never left either on or off – therefore the lamp cannot be on at t₁. And then the same reasoning shows that it cannot be off at t₁ either. Yet it must be either on or off. This is a contradiction.

This fact has nothing to do with one's interpretation of mathematics (and nothing to do with the limit of some proposed infinite sequence of numbers).

Are you arguing that Thompson's sequence is finishable hypothetically, but without possessing a definite end value? — sime

No, I'm saying that it isn't completable, even hypothetically. The fact that there is no definite end value is just one way to demonstrate that it isn't completable; the lamp must be either on or off after two minutes, but if the button has been pushed an infinite number of times before then then it cannot be either.

In which case your argument would be closer to constructive mathematics based on intuitionistic logic, rather than to intuitionism. — sime

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use $e$ and $i$, because why not?

The logic of the lamp just has nothing to do with some sequence of additions and subtractions. The code here properly demonstrates the logic.

Excellent observation. What Witt would do is create a situation and give examples of what we’d say. “I’m in pain” “Me too” “But I have a headache.” “Me too!” “Mine’s a shooting zing behind my ear” “Right! Boy, I know your pain.” Thus why he will conclude that, as a matter of identity, to the extent we agree, we have the same pain (PI # 235). — Antony Nickles

The fact that we use the same word "pain" to refer to your sensations and to my sensations isn't that your sensations are the same as my sensations.

I don't think it has anything to do with formalism or intuitionism or anything like that.

Our starting premises are:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t₀
P4. The lamp is either on or off at t₁

Thomson asks the following question:

Q1. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

Compare with:

Q2. If the button is pushed an odd number of times between t₀ and t₁ then is the lamp on or off at t₁?
Q3. If the button is pushed an even number of times between t₀ and t₁ then is the lamp on or off at t₁?

Benacerraf claims that we can simply stipulate that the lamp is on at t₁.

There are two problems with this.

The first is that we cannot simply stipulate the answer. I cannot stipulate that the answer to Q2 is "off" and I cannot stipulate that the answer to Q3 is "on". The answers must be deducible from the premises.

The second problem is that the following is deducible from the first three premises:

C1. If the button is pushed an infinite number of times between t₀ and t₁ then the lamp is neither on nor off at t₁

This is because if the button is pushed at least once then for the lamp to be on at t₁ the button must have been pushed to turn and leave it on for t₁, and for the lamp to be off at t₁ the button must have been pushed to turn and leave it off for t₁, neither of which are possible if the button is pushed an infinite number of times between t₀ and t₁.

C1 contradicts P4.

Given that P4 is necessarily true, it follows that the antecedent of C1 is necessarily false. So it is metaphysically impossible to have pushed the button an infinite number of times.